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\begin{document}

\title{Price and greeks of European binary option}
\date{}

\maketitle


Consider an asset with spot price $S = \{S_t; 0 \leq t \leq T\}$ following geometric Brownian motion of volatility $\sigma$.

A European binary call option with strike $K$ and maturity $T$ pays off
\begin{align}
    \text{Payoff}_\text{Call}
        = 1_{S_T \geq K} ,
\end{align}
where
$1_{S_T \geq K}$ is an indicator function.
A European binary put option with the same strike and maturity pays off
\begin{align}
    \text{Payoff}_\text{Put}
        = 1_{S_T \leq K} .
\end{align}



\section*{Price}


The price of the European binary call option is given by
\begin{align}
    \text{Price}_\text{Call}
        = \Ex[1_{S_T \geq K}] = N(d_2) ,
\end{align}
where
$N$ is the cumulative distribution function of the normal distribution and
\begin{align}
    d_2
        = \frac{\log (S_0 / K)}{\sigma \sqrt{T}} - \frac12 \sigma \sqrt{T} .
\end{align}

The price of a European binary put option is given by $1 - \text{Price}_\text{Call}$
because a relation $\text{Payoff}_\text{Call} + \text{Payoff}_\text{Put} = 1$ holds almost surely.


\section*{Delta}


Delta is given by
\begin{align}
    \text{Delta}_\text{Call}
        = \frac{N^\prime(d_2)}{S_0 \sigma \sqrt{T}} ,
\end{align}
where
we used a derivative $\partial d_2 / \partial S_0 = 1 / (S_0 \sigma \sqrt{T})$.

Delta of a European binary put option is $\text{Delta}_\text{Put} = - \text{Delta}_\text{Call}$.


\section*{Gamma}


Gamma of the European binary option is given by
\begin{align}
    \text{Gamma}
        = \frac{N^{\prime\prime}(d_2)}{S_0^2 \sigma^2 T}
            - \frac{N^{\prime}(d_2)}{S_0^2 \sigma \sqrt{T}}
        = - \frac{N^{\prime}(d_2)}{S_0^2 \sigma \sqrt{T}}
            \br{\frac{d_2}{\sigma \sqrt{T}} + 1} ,
    \label{eq:gamma}
\end{align}
where we used a relation $N^{\prime\prime}(x) = - x N^\prime(x)$ to show the second equality.

Gamma of a European binary put option is $\text{Gamma}_\text{Put} = - \text{Gamma}_\text{Call}$.


\end{document}
